# Higher Order Derivatives of Laplace Transform/Examples/Example 1

## Examples of Use of Laplace Transform of Integral

Let $\laptrans f$ denote the Laplace transform of the real function $f$.

$\laptrans {t^2 e^{2 t} } = \dfrac 2 {\paren {s - 2}^3}$

## Proof

 $\ds \paren {-1}^2 \laptrans {t^2 e^{2 t} }$ $=$ $\ds \dfrac {\d^2} {\d s^2} \laptrans {e^{2 t} }$ Laplace Transform of Integral $\ds \leadsto \ \$ $\ds \laptrans {t^2 e^{2 t} }$ $=$ $\ds \dfrac {\d^2} {\d s^2} \dfrac 1 {s - 2}$ Laplace Transform of Exponential $\ds$ $=$ $\ds \dfrac 2 {\paren {s - 2}^3}$ simplification

$\blacksquare$